372 XV NORMED ALGEBRAS AND SPECTRAL THEORY follows immediately from the fact that %(x) ^ 0 implies %(x2) ^ 0. The corre- sponding representation U^ is evidently irreducible. We denote by H(A) the set of hermitian characters of A; it is a subset of the product space CA and is closed in the product topology (3.15.1). We give H(A) the topology induced by the product topology (i.e., the weak topology (1 2.1 5)). When A is a separable commutative Banach algebra with involution, having a unit element e 7^ 0, the space H(A) is a compact metrizable subspace of X(A) (15.3.2). In this situation it can happen that H(A) =£ X(A) (Section 15.4, Problem 3), but H(A) is equal to X(A) if A is a separable star algebra (15.4.14). We shall show that the bitraces g satisfying the conditions at the begin- ning of this section can all be obtained, by a canonical process of " integra- tion", from hermitian characters: (1 5.9.2) (Plancherel-Godement theorem) Let g be a bitrace on a com- mutative algebra A with involution, satisfying (U) and (N), and such that the prehilbert space A/ng and the star algebra &$ g a Jzf (H^) are separable. (I) We can define canonically: (1) a subspace S^ <9/H(A), whose closure in CA fs either Sg or Sgu {0} and is metrizable and compact (so that S^ is locally compact, metrizable and separable) ; (2) a positive measure mg on Sg , with the following properties : (i) For each x e A, the function %*-+x(i) = %(x) belongs to &c(Sg, mg) and we have mg(x) (15.9.2.1) g(x, y) = f fay*) dmg(X) - f Js* Js* for all x, y in A. (ii) As x runs through A, the set of functions x is contained in #c(Sfl) (13.20.6) and is dense in this Banach space. (iii) The support of the measure mg is the whole ofSg . (iv) The mapping x\-+x factor izes into x\-~*ng(x)&x, and the mapping T0 ofA/ng into #2(8^) extends to an isomorphism T of the Hilbert space Hg onto L|(S, , mg) such that for allxeA we have Ug(x) = T~lM(x)T, where M(x) is multiplication by the class ofx in L%($g,mg) (cf. (13.21.5)). Also for all 1 e C and x e A. (II) Conversely, let She a subspace of H(A) such that S u {0} is compact and metrizable, and let m be a positive measure on S, with support equal to S,